Definition:P-adic Valuation/P-adic Numbers/Definition 2

Definition

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

The $p$-adic valuation on $p$-adic numbers is the mapping $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ defined by:

$(1): \quad \map {\nu_p} 0 = +\infty$

$(2): \quad $for all $x \in \Q_p \setminus \set 0$:

$\map {\nu_p} x$ is the index of the first non-zero coefficient in the canonical $p$-adic expansion of $x$

Also see

• Equivalence of Definitions of P-adic Valuation on P-adic Numbers

• P-adic Valuation Extends to P-adic Numbers where it is shown that $\nu_p$ is a valuation that extends the $p$-adic valuation on the rational numbers $\Q$ to $\Q_p$.

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$